1D Groundstate wavefunction always even for even potential?

Wavelet
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Hi!
I have calculated various eigenstate wavefunctions for a one-dimensional system of a particle in a potential. The potential is an even function.
All the wavefunctions have become either even or odd functions which I understand why. The ground-state wavefunction has always been even, is this always the case and if so why? If not does anyone know of a system with odd ground-state wavefunction?

Thanks in advance,
Wavelet
 
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For a hamiltonian of the form H = p^2/2m + V(x) in one dimension, the ground state wave function has no nodes (that is, is never zero). So, if it must be odd or even (as it must if V is even), then it has to be even.

I don't have time now to type in a proof that the ground state wave function has no nodes, but it shouldn't be hard to find one on the web somewhere.
 
Thanks!
 
Hi Avodyne,

Good whatever time of day it is where you are.

I'm searching for a rigorous proof of the fact that for any even potential the ground state wave function is always even. Non-relativistic 1 dimensional case proof would be fine for me. Landau Lifschitz, volume #3 the next paragraph after (21.8) equation, produces a proof (if it can be called a proof), but it's not enough for example for V(x)=A*x^4-B*x^2 case.

So could you direct me in this search? A book or web-site?

With best regards,
Anton
 
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